While that was down 16,000 from a revised 430,000 initial claims filed the week before, it was still at a level economists often say is too high to bring down the unemployment rate.

In order to significantly bring down unemployment, economists say the country needs to create at least 200,000 jobs a month or more. And that kind of job growth often is consistent with initial claims below 400,000 said Tim Quinlan, economist with Wells Fargo.

The bad news is initial claims have stayed above that level since early April. The good news is, they trended down last week, he said.

"It is decent news. I don't think it's time to celebrate the return of big job growth or anything, but it's definitely a step in the right direction," he said.

The four-week moving average of initial claims, calculated to smooth out volatility, totaled 424,750, the same as the week before.

Continuing claims -- which include people filing for the second week of benefits or more -- fell 21,000 to 3,675,000 in the week ended June 4, the most recent week available.

Much of the reduction in continuing claims could be due to people running out of unemployment benefits altogether, which last for as long as 22 months in some states According to the Labor Department, about 6.2 million people have been unemployed for more than six months.